The Kneser--Poulsen conjecture for special contractions

Károly Bezdek, Márton Naszódi

Published 2017-01-18Version 1

The Kneser--Poulsen Conjecture states that if the centers of a family of $N$ balls in ${\mathbb E}^d$ is contracted, then the volume of the union (resp., intersection) does not increase (resp., decrease). We consider two types of special contractions. First, a \emph{uniform contraction} is a contraction where all the pairwise distances in the first set of points are larger than all the pairwise distances in the second set of points. For balls of equal radii, we obtain that a uniform contraction of the centers does not decrease the volume of the intersection of the balls, provided that $N\geq(1+\sqrt{2})^d$. We prove a similar result concerning the volume of the union. Second, a \emph{strong contraction} is a contraction in each coordinate. We show that the conjecture holds for strong contractions. In fact, the result extends to arbitrary unconditional bodies in the place of balls.